Wilson Loops and Chiral Correlators on Squashed Sphere

Abstract

We study chiral deformations of ${\cal N}=2$ and ${\cal N}=4$ supersymmetric gauge theories obtained by turning on $τ_J \,{\rm tr} \, Φ^J$ interactions with $Φ$ the ${\cal N}=2$ superfield. Using localization, we compute the deformed gauge theory partition function $Z(\vecτ|q)$ and the expectation value of circular Wilson loops $W$ on a squashed four-sphere. In the case of the deformed ${\cal N}=4$ theory, exact formulas for $Z$ and $W$ are derived in terms of an underlying $U(N)$ interacting matrix model replacing the free Gaussian model describing the ${\cal N}=4$ theory. Using the AGT correspondence, the $τ_J$-deformations are related to the insertions of commuting integrals of motion in the four-point CFT correlator and chiral correlators are expressed as $τ$-derivatives of the gauge theory partition function on a finite $Ω$-background. In the so called Nekrasov-Shatashvili limit, the entire ring of chiral relations is extracted from the $ε$-deformed Seiberg-Witten curve. As a byproduct of our analysis we show that $SU(2)$ gauge theories on rational $Ω$-backgrounds are dual to CFT minimal models.